View of Numerical Invariants of Coprime Graph of A Generalized Quaternion Group

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Vol. 29, No. 01 (2023), pp. 36–44.

NUMERICAL INVARIANTS OF COPRIME GRAPH OF A GENERALIZED QUATERNION GROUP

Nurhabibah^1,a, I Gede Adhitya Wisnu Wardhana^1,b, and Ni Wayan Switrayni^1,c

1Department of Mathematics, Faculty of Mathematics and Natural Science, University of Mataram. Email:

bcorresponding author: [email protected]

Abstract.The coprime graph of a finite group was defined by Ma, denoted by ΓG, is a graph with vertices that are all elements of groupGand two distinct verticesx andyare adjacent if and only if (|x|,|y|) = 1. In this study, we discuss the numerical invariants of a generalized quaternion group. The numerical invariant is a property of a graph in numerical value and that value is always the same on an isomorphic graph. This research is fundamental research and analysis based on patterns in some examples. Some results of this research are the independence number of ΓQ_4n

is 4n−1 or 3nand its complement metric dimension is 4n−2 for eachn≥2.

Key words: coprime graph, generalized quaternion group, numerical invariants

1. INTRODUCTION

The graph representation of an algebraic structure becomes a hot topic in recent years. For a finite groupG, we can representGin some simple graphs such as the coprime graph, the non-coprime graph, the power graph, the intersection graph, the commuting graph, and others. See [1] [2] [4] [11] [12] for details. In 2014, Ma et al [1] introduced the coprime graph of a finite group G, where the vertex set of the graph is Gand two distinct vertices xandy are adjacent if and only if the order of xand the order ofy are relative primes [1].

In 2021, Nurhabibah et al. [4] give some results on the shape of the coprime graph of a generalized quaternion group as a bipartite graph, tripartite graph, or multipartite graph. So, in this article, we would like to discuss numerical invariants

2020 Mathematics Subject Classification: 05C25 Received: 04-07-2022, accepted: 22-02-2023.

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of the graph as degree, radius, diameter, domination number, independence number, girth, metric dimension, and complement metric dimension. The numerical invariant is a property of a graph in numerical value and that value is always the same on an isomorphic graph.

2. DEFINITION AND SOME PROPERTIES

A generalized quaternion group is a special group with a definition as follows.

Definition 2.1. [5] A Generalized quaternion group (Q_4n)with n≥2 is a group with a presentation

< a, b|a²ⁿ =e, aⁿ=b², b⁻¹ab=a⁻¹>

in this group, a^kb=ba^−k and the order ofa^kbis 4.

For example,D₆={e, a, a², b, ab, a²b}is a dihedral group with order six. In this study, we give a functionf that defines the adjacency of two vertices as follows.

Definition 2.2. GivenH = (V, E)is a graph withV 6=∅. Define f as follows.

f :V ×V → {0,1}

with

f((vi, vj))

1, if(vi, vj)∈E

0, else

One graph that is associated with a finite group is the coprime graph. The definition of this graph is given in Definition 2.3.

Definition 2.3. [1] Given G a finite group, the coprime graph of G, denoted by Γ_G is the graph with V(Γ_G) =Gand two distinct verticesx andy are adjacent if and only if(|x|,|y|) = 1.

And now, we define some numerical invariants of the graph that we analyze in this study.

Definition 2.4. Let G be a graph. The degree of vertex v in G is denoted as deg(v) =|A| whereA={vi∈V|f((v, v_i)) = 1}.

Definition 2.5. [6] Let G be a graph and c(v) = max{d(v, x)|x ∈ V(G)} is the eccentricity of G, rad(G) =min{c(v)|v ∈V(G)} and diam(G) = maks{c(v)|v ∈ V(G)}.

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Definition 2.6. [7]Let G= (V, E) be a graph. The girth of G, denoted byg(G) is defined as:

g(G) =

min{ |CG||C_G cycle ofG}, Gcontains cycle

0, else

Where|CG| is the length of the cycleCG.

Definition 2.7. [8] Let G be a graph, the domination number of G, denoted by γ(G) is the minimum cardinality of the domination set ofG. The domination set is a subset D of V(G) such that every vertex not inD is adjacent to at least one member of D.

Definition 2.8. [9] Let Gbe a graph, the independence number of G, denoted by β(G)is the maximum cardinality of the independence set ofG. The independence set is a subset I of V(G) such that no two vertices in the subset represent an edge of V(G).

Definition 2.9. [10] Let G be a graph, the metric dimension of G, denoted by dim(G)is the minimum cardinality of the resolving set of G. The resolving set is a subset W for a graph G if, for every two distinct verticesuand v of G, there is an element win W that resolves uandv.

Definition 2.10. [11] Let G be a graph, complement metric dimension ofG, denoted by dim(G) is the maximum cardinality of complement resolving a set ofG.

Definition 2.11. LetΓGbe the coprime graph of a finite groupGandx∈V(ΓG).

We can defineAxas Ax={xi∈V(ΓG)|f((x, xi)) = 1}.

We will discuss numerical invariants of the generalized quaternion group that is represented in the coprime graph. Given the previous result of the coprime graph of a generalized quaternion group.

Theorem 2.12. [4]Let Q4n be a generalized quaternion group. Ifn= 2^k then the coprime graph ofQ4n is a complete bipartite graph.

Theorem 2.13. [4]LetQ4n be a generalized quaternion group. Ifnis prime, then the coprime graph of Q4n is tripartite graph.

Theorem 2.14. [4]LetQ4n be a generalized quaternion group. Ifn=p^k₁¹p^k₂²...p^k_m^m, pi6= 2, pi 6=pj for each i6=j, pi a prime number then the coprime graph ofQ4n is m+ 2partite graph.

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Theorem 2.15. [4]LetQ_4n be a generalized quaternion group. Ifn=p^k₁¹p^k₂²...p^k_m^m, p1= 2,pi are distinct prime numbers then the coprime graph ofQ4nism+1partite graph.

We also give the following theorem that we need to prove some numerical invariants of the coprime graph of a finite group.

Theorem 2.16. Let (G,∗) be a finite group, |G| = n =p^k₁¹p^k₂²...p^k_m^m and ΓG be the coprime graph of G. For some x, y ∈ V(ΓG) with x 6= y, 1 ≤ li, ti ≤ ki,

|x|=Qj

i=1p^l_iⁱ and|y|=Qj

i=1p^t_iⁱ if and only if A_x=A_y.

Proof. Take any x1 ∈ Ax, it means (|x1|,|x|) = 1. Note that |x| = Qj i=1p^l_iⁱ and (|x1|,|x|) = 1, so 6 ∃pi with 1 ≤i ≤j such that pi||x1|. Since |y|=Qj

i=1p^t_iⁱ and 6 ∃pi with 1 ≤i ≤ j which resulted in pi||x1| then (|y|,|x1|) = 1 and hence f((x1, y)) = 1. Thus,x1∈Ay, which means Ax⊆Ay. By a similar approach, we can prove thatAy ⊆Ax, henceAx=Ay

LetGbe a finite group with|G|=n=p^k₁¹p^k₂²...p^k_m^m and ΓG be the coprime graph of G. Define A = {1,2, ..., m} and B, C ⊆ A with B 6= C. If x, y ∈ G,

|x|=Q

s∈Bp^l_s^s and|y| =Q

r∈Cp^l_r^r , we need to prove A_x 6=A_y. Choosez ∈ A_x with|z|=p_a wherea∈C, buta /∈B. Consequently, (|y|,|z|) =p_a6= 1. It means z /∈A_y. So, it is complete to prove A_x6=A_y.

3. NUMERICAL INVARIANTS OF THE COPRIME GRAPH OF Q_4n

In this section, we analyze the numerical invariants of the coprime graph of Q_4n. The first numerical invariants obtained from this study are the degree of each vertex as stated in Theorem 3.1 and Theorem 3.2.

Theorem 3.1. Let ΓQ_4n be the coprime graph of Q4n. Ifn= 2^k withk≥1 then deg(e) = 4n−1 anddeg(v) = 1for each v∈Q4n\{e}.

Proof. Let Γ_Q_4n be the coprime graph ofQ_4n. Taken= 2^k. By Theorem 2.12, Γ_Q_4nis a complete bipartite graph with partitionsV₁={e}andV₂=Q_4n\{e}, see the proof in [4] for detail. Thus,f((e, v)) = 1, for eachv∈Q_4n\{e}. By definition, obtained deg(e) =|V₂|= 4n−1 anddeg(v) =|V₁|= 1.

And fornis an odd prime number we have

Theorem 3.2. LetΓQ_4nbe the coprime graph ofQ4n. Ifnis an odd prime number, the degree of each vertex is 1, n,2n+ 2, or4n−1.

Proof. Let Γ_Q_4n be the coprime graph ofQ_4n. Takenfor any odd prime number p. Let S₁ = {e}, S₂ = {a^p, b, ab, ..., a^2p−1b}, S₃ = {a², a⁴, ..., a^2p−2}, and S₄ = {a, a³, ..., a^p−2, a^p+2, ..., a^2p−1}. Note that |e| = 1, |x| = 2 or |x| = 4 for each x∈S₂,|y|=pfor eachy∈S₃, and|z|= 2pfor eachz∈S₄. Thus,

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• f((e, v₁)) = 1 for eachv₁∈A₁ withA₁=Q_4n\{e}, sodeg(e) =|A₁|= 4n−1.

• Forx∈S2, we getf((x, v2)) = 1 if and only if v2∈A2 withA2=S1∪S3. So, deg(x) =|A2|=|S1|+|S3|=n.

• Fory ∈S3, we get f((y, v3)) = 1 if and only if v∈A3 withA3=S1∪S2. So, deg(y) =|A3|=|S1|+|S2|= 1 + 2n+ 1 = 2n+ 2.

• For z ∈ S4, we get f((z, v4)) = 1 if and only if v4 ∈ A4 with A4 = S1. So, deg(z) =|S1|= 1.

The next numerical invariants are radius and diameter.

Theorem 3.3. IfΓ_Q_4n is the coprime graph of a generalized quaternion group with n≥2, thenrad(Γ_Q_4n) = 1anddiam(Γ_Q_4n) = 2.

Proof. Let Γ_Q_4n be the coprime graph ofQ_4n. Take anyn∈Nandn≥2. From [4] we know that 1≤ d(u, v)≤2 for each u, v ∈ V(Γ_Q_4n). It meansc(v) = 1 or c(v) = 2 and by definition we get rad(Γ_Q_4n) = 1 anddiam(Γ_Q_4n) = 2.

The next result obtained from this study is the girth of ΓQ_4nas stated in the two following theorem.

Theorem 3.4. IfΓ_Q_4n is the coprime graph of a generalized quaternion group with n= 2^k theng(ΓQ_4n) = 0.

Proof. According to Theorem 2.12, if n= 2^k then Γ_Q_4n is a complete bipartite graph. It means, there is no cycle in Γ_Q_4n. By definition, we getg(Γ_Q_4n) = 0.

And for other cases we have

Theorem 3.5. IfΓQ_4n is the coprime graph of a generalized quaternion group with n6= 2^k andn >2 theng(ΓQ_4n) = 3.

Proof. Let ΓQ4n is the coprime graph of a generalized quaternion group Q4n. Take n ∈ Nwith n > 2 and n 6= 2^k. There arev1, v2 ∈V (ΓQ_4n) with|v1|= p, p an odd prime number and |v2| = 2. So, we get a cycle with length 3 which is e−v₁−v₂−v₁. It is complete to proveg(Γ_Q_4n) = 3.

The domination number is one of the numerical invariants of the graph. The domination number of ΓQ_4n is always the same for all n ≥ 2. This property is stated in Theorem 3.6.

Theorem 3.6. If ΓQ_4n be the coprime graph of a generalized quaternion then γ(ΓQ_4n) = 1for each n≥2.

Proof. LetD={e} andD be the domination set because eis adjacent to other vertices in ΓQ4n. So,γ(ΓQ4n) =|D|= 1.

The next numerical invariant is the independence number of the graph. This property is stated in four following Theorem.

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Theorem 3.7. Let Γ_Q_4n be the coprime graph of a generalized quaternion group.

If n= 2^k thenβ(ΓQ_4n) = 4n−1.

Proof. Let ΓQ_4n be the coprime graph of a generalized quaternion group with n = 2^k. By Theorem 2.12, Γ_Q_4n is a complete bipartite graph that is a star graph. It means there are two partitions and these partitions are independent sets, V₁ = {e} and V₂ = Q_4n\{e}. Now, suppose that there is independence set I with |Q4n| ≥ |I| > |V2|. So, I must equal to Q4n and f((e, v)) = 1 for any v ∈ Q4n\{e}. It is a contradiction and it means V2 is an independence set with maximum cardinality. Finally, we getβ(Q4n) =|V2|= 4n−1.

And for casenis an odd prime we have

Theorem 3.8. Let Q4n be a generalized quaternion group, n=p,pan odd prime number, thenβ(ΓQ_4n) = 3n.

Proof. Let ΓQ_4nbe the coprime graph ofQ4n. Wherenis an arbitrary odd prime number. Define I = {a^j|j = 1,3, ...,2n−1} ∪ {aⁱb|i = 0,1,2, ...,2n−1}. Note that |v|= 2q, q∈N,∀v∈I. Thus, ∀vi, vj ∈I it is valid that (|vi|,|vj|)6= 1 which resulted inf((vi, vj)) = 0. So,I is the independence set of ΓQ_4n. And we need to prove that there is no independence set of ΓQ_4n with condition|I⁰|>|I|= 3n. If there isI⁰, we can findv∈I⁰, butv /∈I, it means|v|= 1 or|v|=p. Consequently, there isu∈I∩I⁰ with condition (|u|,|v|) = 1 which means f((u, v)) = 1. It is a contradiction withI⁰ independence set of ΓQ_4n. So,β(ΓQ_4n) =|I|= 3n.

And for a more general case, whennis an odd

Theorem 3.9. LetΓQ_4nbe the coprime graph ofQ4n. Ifn=p^k₁¹p^k₂²...p^k_m^m,pi6=pj, i6=j,pi an odd prime number, thenβ(ΓQ_4n) = 3n.

Proof. Let ΓQ_4nbe the coprime graph ofQ4n. Taken=p^k₁¹p^k₂²...p^k_m^m,pi6= 2,pi6=

p_j,i6=j,p_ian odd prime number. DefineI=I₁∪I₂withI₁={a^2l−1|l= 1,2, ..., n}

and I2 = {aⁱb|i = 0,1,2, ...,2n−1}. Take a^2l−1 ∈ I1, we get |a^2l−1| = 2z1 with z1 ∈N and∀aⁱb∈I2 we get |aⁱb|= 4. Thus,∀u, v ∈I we get (|u|,|v|) = 2z2 6= 1 which shows that f((u, v)) = 0. So,I is an independence set. Suppose that I⁰ is an independence set with|I⁰|>|I|= 3n. We can findv₁∈I⁰ andv₁∈/I, it means

|v1| 6= 2z3, ∀z3 ∈Z. Because of |I⁰| >3n then |I∩I⁰| ≥ 2n+ 1. Consequently,

∃v₂ ∈ I∩I⁰ with v₂ = aⁱb with i ∈ {0,1, ...,2n−1}, it means |v₂| = 4. Thus, (|v₁|,|v₂|) = 1 orf((v₁, v₂)) = 1. It is a contradiction withI⁰ is independence set.

So,β(Γ_Q_4n) =|I|= 3n.

And for a more general result

Theorem 3.10. IfΓ_Q_4n be the coprime graph ofQ_4n. Ifn=p^k₁¹p^k₂²...p^k_m^m,p₁= 2, p_i6=p_j,i6=j thenβ(Γ_Q_4n) = 4n−₂ⁿk1.

Proof. Let ΓQ_4n be the coprime graph of Q4n. Taken= p^k₁¹p^k₂²...p^k_m^m, p1 = 2, p_i6=p_j,i6=j. DefineI=I₁∪I₂∪I₃withI₁={a^2l−1|l= 1,2, ..., n},I₂={a^2l|l=

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1,2, ..., n−1}\{a²^k^{1 +1}^q|q = 1,2, ...,₂ⁿ_k₁ −1}, and I₃ ={aⁱb|i= 0,1,2, ...,2n−1}.

For any v∈I we get|v|= 2z1, z1∈N. Consequently,∀u, v ∈I we get (|u|,|v|) = 2z26= 1. ThusI is an independence set. And now, we need to prove that there is noI⁰ as the independence set of ΓQ_4nwith condition|I⁰|>|I|. Suppose that there is I⁰ with this condition, then∃v1∈I⁰ andv1 ∈/ I, It means|v1| 6= 2z3, ∀z3 ∈Z.

Because of|I⁰|>|I|then|I∩I⁰| ≥4n−₂kⁿ1−1+ 1>3n. Consequently,∃v2∈I∩I⁰ wherev2=aⁱband|v2|= 4. Those, we get (|v1|,|v2|) = 1 orf((v1, v2)) = 1. It is a contradiction. So,β(ΓQ_4n) =|I|=|I1|+|I2|+|I3|=n+ (n−1)− ₂ⁿ_k₁ −1

+ 2n= 4n−₂ⁿk1.

Two of the numerical invariants of the graph that are defined based on the metric concept are the metric dimension and the complement metric dimension.

In this study, we find the metric dimension and complement metric dimension of Γ_Q_4n.

Theorem 3.11. LetΓQ_4nbe the coprime graph ofQ4n. Ifn= 2^k thendim(ΓQ_4n) = 4n−2.

Proof. Let Γ_Q_4n be the coprime graph of Q_4n. If n = 2^k, k ∈ N, then Γ_Q_4n is star graph. Choose W = Q4n\{e, a} = {v1, v2, ..., v_4n−2},∀v ∈ W obtained r(vi|W) = (d1, d2, ..., d_4n−2) with di = 0 and dj = 2 for each i 6= j. So, each v ∈ W has a distinct representation of W. And then r(e|W) = (1,1, ...,1) and r(a|W) = (2,2, ...,2). Thus, each vertex of ΓQ_4n has distinct representation toW. In other words,W is a resolving set of ΓQ_4n.

Suppose that there isW⁰ as a resolving set with condition |W⁰| < 4n−2.

There are two distinct vertices v1, v2 ∈ Q4n\{e} and v1, v2 ∈/ W⁰ which resulted in r(v1|W⁰) = r(v2|W⁰). Thus, W⁰ is not resolving set and it is complete to dim(ΓQ_4n) =|W|= 4n−2.

And fornis an odd prime number

Theorem 3.12. Let ΓQ_4n be the coprime graph of Q4n. Ifn=p,pan odd prime number, thendim(ΓQ_4n) = 4n−4.

Proof. Let ΓQ_4n be the coprime graph ofQ4n. Letnbe any odd prime number.

Choose W =Q4n\{e, a, a², b}. Note that eachv1, v2 ∈W with v1 6=v2 obtained r(v1|W)6=r(v2|W). Because of|e| 6=|a| 6=|a²| 6=|b|and the set of prime numbers that divide the order ofe, a, a², andb are not equal, so by Theorem 2.16 obtained Ae 6=Aa 6=A_a2 6=Ab. So, r(e|W)6=r(a|W)6=r(a²|W)6=r(b|W). Thus W is a resolving set of ΓQ_4n.

Suppose that there isW⁰ with condition|W⁰|<|W|= 4n−4 as a resolving set of ΓQ_4n. We can find two distinct vertices u, v /∈ W⁰ with |u| = |v|, means Au = Av and r(u|W⁰) = r(v|W⁰). Thus, W is a resolving set with minimum cardinality. In other words,|W”| ≥4n−4 for eachW” resolving set of ΓQ4n. So, dim(ΓQ4n) =|W|= 4n−4.

And fornis an odd number

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Theorem 3.13. Let Γ_Q_4n be the coprime graph ofQ_4n. Ifn=p^k₁¹p^k₂²...p^k_m^m,p_i an odd prime number,pi6=pj fori6=j then dim(ΓQ_4n) = 4n−2^m+1.

Proof. Let Γ_Q_4n be the coprime graph of Q_4n. Take n = p^k₁¹p^k₂²...p^k_m^m, p_i 6=

pj for i 6= j. Define W = Q4n\S with S = {e} ∪S1∪...∪Sm+1 and St = {vt1, vt2, ..., v_tCm+1

t } for each 1 ≤t≤m+ 1 with |vts| has t distinct prime factor and there are no two distinct elements inS_twithtthe same prime factor. Thus,|St| is the total of ways to choosetprime numbers fromm+ 1 distinct prime numbers or|St|=C_t^m+1 where C_k^m=_(m−k)!k!^m! . Consequently,

|S|= 1 +C₁^m+1+C₂^m+1+...+C_m+1^m+1= 2^m+1

Suppose that W⁰ = Q4n\S⁰ with |S⁰| > |S| is resolving set of ΓQ_4n. We can find v5, v6 ∈S⁰ withv56=v6, but |v5| and |v6| have t the same prime factor.

By Theorem 2.16, obtained Av₅ =Av₆ and r(v5|W⁰) = r(v6|W⁰). So, W⁰ is not resolving set andW is resolving set ΓQ4n with minimum cardinality. It is done to provedim(Γ_Q_4n) =|W|= 4n−2^m+1.

And for a more general case, by Theorem 3.13 we have the following result.

Corollary 3.14. Let Γ_Q_4n be the coprime graph of Q_4n. If n = p^k₁¹p^k₂²...p^k_m^m, p₁= 2,p_i6=p_j fori6=j,p_i prime number, thendim(Γ_Q_4n) = 4n−2^m.

Proof. Let Γ_Q_4n be the coprime graph of Q_4n. Taken= p^k₁¹p^k₂²...p^k_m^m, p₁ = 2, p_i 6= p_j for i 6= j, p_i prime number. Note that p₁ = 2, so the number of odd prime number p_j is m−1. Consequently, Theorem 3.13 obtained dim(Γ_Q_4n) = 4n−2^(m−1)+1= 4n−2^m.

The last theorem explains the complement metric dimension of ΓQ_4n.

Theorem 3.15. IfΓ_Q_4nbe the coprime graph ofQ_4nwithn≥2thendim(Γ_Q_4n) = 4n−2.

Proof. Let Γ_Q_4n be the coprime graph of Q_4n. Choose any natural number n with n ≥ 2. Choose S = Q_4n\{b, ab}, obtained r(b|S) = r(ab|S). So, S is a complement to resolving a set of Q_4n. By definition, it is impossible to find a complement resolving set with cardinality greater than |V(G)| −2. Thus, S is a complement resolving set with maximum cardinality. It is done to prove that dim(ΓQ_4n) =|S|= 4n−2.

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4. CONCLUSION

Some numerical invariants of the coprime graph of a generalized quaternion that were obtained from this study are the degree of each vertex is 1, n,2n+ 2, or 4n−1, its radius is 1, its diameter is 2, its girth is 0 or 3, its independence number is 4n−1, 3n, or 4n−₂ⁿ_k₁, its domination number is 1, its metric dimension is 4n−2,4n−4,4n−2^m, or 4n−2^m+1, and its complement metric dimension is 4n−2.

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